Forums > Pricing & Modelling > closed form pdf of time series predictions

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 nikol Total Posts: 795 Joined: Jun 2005
 Posted: 2018-11-12 13:16 I am looking for 'econometric' models which deliver closed-form solution for predicted p.d.f.. For example, ARMA(1,1) with normal innovations delivers prediction following Normal pdf. Distribution expressed as finite decomposition of 'famous' distributions such as Normal, t-Student, Binomial, Gamma, X2 etc. is also acceptable. Should admit that although I used quite a number of "econometric" models I still miss high level view on this subject, because never liked this 'polynomial' approach.If you know the list from top of your head, please, share. So far, I found this "Probabilistic time series forecasting with boosted additive models" , but that's quite not what I want...
 ronin Total Posts: 488 Joined: May 2006
 Posted: 2018-11-27 12:36 The general approach to that sort of thing is to write the Fokker-Planck equation for your dynamics, and then solve it for the diffusion of the delta function. BTW that's also how we know what the normal distribution looks like.Generally, you won't be able to write a simple formula for the pdf. PDEs don't work like that. But you can still solve them numerically or asymptotically. "There is a SIX am?" -- Arthur
 nikol Total Posts: 795 Joined: Jun 2005
 Posted: 2018-11-28 15:12 I would love to avoid numericals as much as I can, since I would like to deliver next prediction within 50-100 microseconds. If unavoidable, have to experiment with what you are saying.
 doomanx Total Posts: 27 Joined: Jul 2018
 Posted: 2018-11-28 16:16 Sounds like you want something Bayesian. Maybe GP regression would be a good fit, although note Bayesian is again usually solved numerically. You can use numerical methods at this time-frame if you are serious about it (this involves a lot of pre-computation and efficient population of the cache).
 ronin Total Posts: 488 Joined: May 2006
 Posted: 2018-11-28 16:41 > next prediction within 50-100 microsecondsThen do the numerics in advance and look them up at decision time. Or do some asymptotics. "There is a SIX am?" -- Arthur
 nikol Total Posts: 795 Joined: Jun 2005
 Posted: 2018-11-28 17:19 Look-up is good.I try to explore recurrent relationships like those in state-dynamics: r_t+1 = (1-w).r_t + w.e_tIt helps to save memory by keeping fewer variables.
 jr Total Posts: 3 Joined: Apr 2017
 Posted: 2018-11-28 21:14 I would have a look at Bayesian filtering as doomanx suggests. Depending on the desired distribution I would choose some type of Kalman filter or designed my own dynamics with particle filters also known as sequential monte carlo.
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